We consider the problem of online linear regression in the stochastic setting. We derive high probability regret bounds for online ridge regression and the forward algorithm. This enables us to compare online regression algorithms more accurately and eliminate assumptions of bounded observations and predictions. Our study advocates for the use of the forward algorithm in lieu of ridge due to its enhanced bounds and robustness to the regularization parameter. Moreover, we explain how to integrate it in algorithms involving linear function approximation to remove a boundedness assumption without deteriorating theoretical bounds. We showcase this modification in linear bandit settings where it yields improved regret bounds. Last, we provide numerical experiments to illustrate our results and endorse our intuitions.

翻译：我们考虑在随机环境下在线线性回归的问题。 我们为在线脊回归和前方算法得出了高概率的遗憾度。 这使我们能够更准确地比较在线回归算法, 并消除对受约束的观察和预测的假设。 我们的研究主张使用远方算法来取代脊, 因为它的边框增强, 且对正规化参数的坚固性。 此外, 我们解释如何将它纳入涉及线性函数近似的算法中, 以便消除线性假设, 而不使理论界限恶化。 我们用线性土匪设置来展示这种修改, 从而产生更好的遗憾界限。 最后, 我们提供数字实验来说明我们的结果, 并认可我们的直觉 。